1. Field of the Invention
The present invention relates to compensation of physical layer impairments on transmitter systems and particularly to a method and apparatus for I/Q imbalance calibration in transmitter systems.
2. Description of the Prior Art
OFDM is a multi-channel modulation system employing Frequency Division Multiplexing (FDM) of orthogonal sub-carriers, each modulating a low bit-rate digital stream. The simplest way to describe an orthogonal frequency-division multiplexing (OFDM) signal is as a set of closely spaced frequency-division multiplexed carriers. While this is a good starting point for those unfamiliar with the technology, it falls short as a model for analyzing the effects of signal impairment.
The reason it falls short is that the carriers are more than closely spaced; they are heavily overlapped. In a perfect OFDM signal, the orthogonality property prevents interference between overlapping carriers. This is different from the FDM systems. In FDM systems, any overlap in the spectrums of adjacent signals will result in interference. In OFDM systems, the carriers will interfere with each other only if there is a loss of orthogonality. So long as orthogonality can be maintained, the carriers can be heavily overlapped, allowing increased spectral efficiency.
Table 1 lists a variety of common analog signal impairments and their effects on both OFDM signals and the more familiar single-carrier modulations such as quadrature phase-shift keying (QPSK) or 64-QAM (quadrature amplitude modulation). Most of these impairments can occur in either the transmitter or the receiver.
ImpairmentOFDMQPSKI/Q gain balanceState spreadingDistortion of(uniform/carrier)constellationI/Q quadrature skewState spreadingDistortion of(uniform/carrier)constellationI/Q channel mismatchState spreadingState spreading(non-uniform/carrier)UncompensatedState spreadingSpinning constellationfrequency errorPhase noiseState spreadingConstellation phase(uniform/carrier)arcingNonlinear distortionState spreadingState spreadingLinear distortionUsually no effectState spreading if not(equalized)equalizedCarrier leakageOffset constellationOffset constellationfor center carrieronly (if used)Frequency errorState spreadingConstellation phasearcingAmplifier droopRadial constellationRadial constellationdistortiondistortionSpuriousState spreading orState spreading,shifting of affectedgenerally circularsub-carrier
IQ Imperfections
For cost reasons, analog in-phase and quadrature (I/Q) modulators and demodulators are often used in transceivers—especially for wide bandwidth signals. Being analog, these I/Q modulators and demodulators usually have imperfections that result in an imperfect match between the two baseband analog signals, I and Q, which represent the complex carrier. For example, gain mismatch might cause the I signal to be slightly smaller than the Q. In a single-carrier modulation system, this results in a visible distortion in the constellation—the square constellation of a 64-QAM signal would become rectangular.
To better understand how gain imbalance will affect an OFDM signal, look at the equations describing each individual sub-carrier. In the following analysis, it's important to keep in mind that, while an individual sub-carrier is analyzed, the IQ gain imbalance error is on the signal that is the composite of all sub-carriers.
In the equation (1), Ck,m is a complex number representing the location of the symbol within the constellation for the Kth sub-carrier at the mth symbol time. For example, if sub-carrier k is binary-phase-shift-keying (BPSK) modulated, then Ck,m might take on values of ±1+j0. The complex exponential portion of Equation 1 represents the Kth sub-carrier, which is amplitude- and phase-modulated by the symbol Ck,m. Therefore:Ck,m(ej2πkΔft)  (1)
Using Euler's relation, the equation (1) can be rewritten as:Ck,m (cos(2π·kΔft)+j sin(2π·kΔft))  (2)
Now add the term “β” to represent gain imbalance. For a perfect signal, set β=0. As shown, the gain imbalance term will also produce a gain change. This was done to simplify the analysis. Therefore:Ck,m((1+β)cos(2π·kΔft)+j sin(2π·kΔft)  (3)
The equation can be rearranged and this can be rewritten as the sum of a perfect signal and an error signal:Ck,m(cos(2π·kΔft)+j sin(2π·kΔft))+Ck,mβ cos(2π·kΔft)  (4)
Finally, converting back into complex exponential notation, we get:
                                          C                          k              ,              m                                ⁢                      ⅇ                          j2π              ⁢                                                          ⁢              k              ⁢                                                          ⁢              Δ              ⁢                                                          ⁢              f              ⁢                                                          ⁢              t                                      +                              (                                          C                                  k                  ,                  m                                            ⁢                              β                2                                      )                    ·                      (                                          ⅇ                                  j2π                  ⁢                                                                          ⁢                  k                  ⁢                                                                          ⁢                  Δ                  ⁢                                                                          ⁢                  ft                                            +                              ⅇ                                                      -                    j2π                                    ⁢                                                                          ⁢                  k                  ⁢                                                                          ⁢                                      Δ                    ⁢                    ft                                                                        )                                              (        5        )            
In words, the equation (5) shows that a gain imbalance produces two error terms. The first error term is at the frequency of the Kth sub-carrier. The second error term is at the frequency of the −Kth sub-carrier. The phase and magnitude of the error terms are proportional to the symbol being transmitted on the Kth sub-carrier. Another way of saying this is that I/Q gain imbalance will result in each sub-carrier being interfered with by its frequency mirror-image sub-carrier. Persons skilled in the art will instantly recognize this as imperfect sideband cancellation.
The equation (5) has several implications. First, it is generally true that for sub-carriers used to carry data (as opposed to pilots), the symbol being transmitted at any given time on the Kth sub-carrier is uncorrelated to the symbol on the −Kth sub-carrier.
For a given sub-carrier, the lack of correlation from the mirror-image sub-carrier implies a certain randomness to the error. This results in a spreading of the sub-carrier's constellation states in a noise-like fashion. This is especially true for higher-order modulations such as 64-QAM. For lower-order modulations, such as BPSK, the error term from the mirror-image carrier has fewer states.
This can result in constellations where the BPSK pilot carriers of an 802.11a signal exhibit spreading that does not appear noise-like. Also, as the BPSK pilots do not have an imaginary component; the error terms associated with the pilot sub-carriers are real—so the spreading is only along the real (I) axis. Note that the phase relationships between the pilot carriers in an 802.11a system are highly correlated, so the errors introduced by quadrature errors are not random.
Quadrature skew produces error terms similar to those produced by gain imbalance. Quadrature skew occurs when the two oscillators used in an I/Q modulator or demodulator do not differ by exactly 90°. For a small angular error, it can be shown that the resulting error is orthogonal to the data. This is indicated by the j in front of the error terms in the equation (6). As with gain imbalance, the error generates energy at the Kth and −Kth sub-carriers. Again, the 802.11a BPSK pilots do not have an imaginary component, so the error term, which is now orthogonal, causes spreading along the Q axis. For the QPSK carriers in this example, the error is also orthogonal. However, unlike BPSK, a QPSK constellation doesn't look any different when rotated by 90°. (See the equation (6).):
                                          C                          k              ,              m                                ⁢                      ⅇ                                          j2π                ⁢                                                                  ⁢                k                ⁢                                                                  ⁢                Δ                ⁢                                                                  ⁢                ft                            ⁢                                                                                  +                  j          ⁢                                                                      C                                      k                    ,                    m                                                  ⁢                ϕ                            2                        ·                          (                                                ⅇ                                      j2π                    ⁢                                                                                  ⁢                    k                    ⁢                                                                                  ⁢                                          Δ                      ⁢                      f                      ⁢                      t                                                                      +                                  ⅇ                                                            -                      j2π                                        ⁢                                                                                  ⁢                    k                    ⁢                                                                                  ⁢                                          Δ                      ⁢                      f                      ⁢                      t                                                                                  )                                                          (        6        )            
In both 802.11a and Hiperlan2, a channel estimation sequence is transmitted at the beginning of a burst. This special sequence is used to train the receiver's equalizer. The intended function of the equalizer is to compensate the received signal for multi-path distortion—a linear impairment in the signal that is the result of multiple signal paths between the transmitter and the receiver. As the ideal channel estimation sequence is known by the receiver, the receiver can observe the effects of the channel on the transmitted signal and compute a set of equalizer coefficients.
In the transmitter, the channel estimation sequence is created by BPSK modulating all 52 carriers for a portion of the preamble. Not coincidentally, the equalizer consists of 52 complex coefficients—one for each sub-carrier. It should come as no surprise that each sub-carrier in the channel estimation sequence has the greatest influence on the equalizer coefficient computed for that same sub-carrier.
The channel estimation sequence, and the receiver algorithms that compute the equalizer coefficients, are not immune from signal impairments. Consider, for example, the effect of I/Q gain imbalance on sub-carriers +26 and −26 of the channel estimation sequence. Recall from Equation 5 that each sub-carrier has two error terms: one at the same frequency as the sub-carrier, and one at the mirror image frequency. The I/Q gain imbalance will cause mutual interference between sub-carriers +26 and −26.
From the IEEE 802.11a standard, the sub-carrier modulation for the channel estimation sequence is defined to be C−26=1+j0 and C+26=1+j0. Using these values in Equation 5, one can easily determine that the two sub-carriers, when combined with the resulting error terms, will suffer an increase in amplitude. The equalizer algorithm will be unable to differentiate the error from the actual channel response, and will interpret this as a channel with too much gain at these two sub-carrier frequencies. The equalizer will incorrectly attempt to compensate by reducing the gain on these sub-carriers for subsequent data symbols.
The result will be different for other sub-carrier pairs, depending on the BPSK channel estimation symbols assigned to each.
With QPSK sub-carriers, the equalizer error caused by gain imbalance, or quadrature skew, results in seven groupings in each corner. Each QPSK sub-carrier suffers from QPSK interference from its mirror image. This results in a spreading to four constellation points in each corner. Each QPSK sub-carrier also suffers from a bi-level gain error introduced by the equalizer. This would produce eight groupings, except that the gain error is such that corners of the groupings overlap at the ideal corner state. Only seven groupings are visible.
IQ Channel Mismatch
When the frequency response of the baseband I and Q channel signal paths are different, an I/Q channel mismatch exists. I/Q channel mismatch can be modeled as a sub-carrier-dependent gain imbalance and quadrature skew. I/Q gain imbalance and quadrature skew, as described above, are simply a degenerate form of I/Q channel mismatch in which the mismatch is constant over all sub-carriers. Think of channel mismatch as gain imbalance and quadrature skew as a function of a sub-carrier. It is still generally true that channel mismatch causes interaction between the Kth and −Kth sub-carriers, but that the magnitude of the impairment could differ between the Kth and the (k+n)th carriers.
Delay mismatch is a distinct error. It can occur when the signal path for the I signal differs in electrical length from the Q signal. This can be caused by different cable lengths (or traces), timing skew between D/A converters used to generate the I and Q signals, or group delay differences in filters in the I and Q signal paths.
What makes this error distinctive is that the error is greater for the outer carriers than it is for the inner carriers. In other words, the error increases with distance from the center sub-carrier. With a vector spectrum display, it may be shown that the signal error is a function of the sub-carrier number (frequency). That is to say, the phase arcing may occur as a function of frequency.
Phase Noise
Phase noise results in each sub-carrier interfering with several other sub-carriers—especially those in close proximity. Close-in phase noise that results in the constellation rotation for the data carriers also results in rotation of the pilot carriers. In fact, carrier phase error rotates all sub-carriers by the same amount, regardless of the sub-carrier frequency. Phase-tracking algorithms use the pilot symbols to detect this common rotation and compensate all of the carriers accordingly. This error is often referred to as common pilot, or common phase error (CPE). Phase noise that is not considered to be close-in results in inter-carrier interference. Instead of constellations with visible rotation, phase noise in an OFDM signal generally results in fuzzy constellation displays, similar to what would be expected if noise is added to the signal.
In 802.11a, the symbol rate is 250 kHz. As pilot symbols are present in every OFDM symbol interval, one might think of this as the sample rate at which the phase-tracking algorithms are operating. Sampling theory would suggest that one will have problems with phase noise beyond one-half the sample rate, or 125 kHz (Nyquist).
Unfortunately, there's another error mechanism that occurs at frequencies much lower than one-half the sample rate. The pilot tracking algorithms are post-fast Fourier transform (FFT). As the FFT is a mapping of the time waveform into the frequency domain (Fourier coefficients), any error that results in energy from one sub-carrier leaking into other sub-carriers cannot be compensated for with a simple post-FFT sub-carrier de-rotation.
Phase noise modulates each of the sub-carriers to the point that they no longer look like simple sinusoids within the FFT interval. Consider an FFT of a single tone. If the tone is pure and has a frequency that produces an integral number of cycles within the FFT time buffer, then an FFT will produce a result with only one non-zero value. If the pure tone is now phase modulated, it can no longer be represented as a single Fourier coefficient—even if the signal is periodic with the FFT time buffer. In other words, the sub-carrier will interfere with the other sub-carriers. If the phase-modulating signal is separated into a DC component plus an AC component, the DC component is corrected by the post-FFT pilot-tracking algorithms.
The impact of random phase noise on the sin(x)/x shape of an individual sub-carrier is a gradual roll-off (an artifact mostly caused by the uniform-windowed FFT used in the receiver). Phase noise causes the nulls of the sin(x)/x spectrum to fill in, creating interference between every sub-carrier and its neighbors.
Frequency Error
In any coherent modulation format, it is critical that the receiver accurately track the transmitter frequency. Frequency is defined to be the derivative of the phase with respect to time, so frequency error can be described as a cumulative phase error that linearly increases or decreases with time, depending on the sign of the frequency error. For single-carrier modulation formats such as 64-QAM, a frequency error can be visualized as a spinning constellation diagram.
The effect of frequency error on OFDM signals is different. Under ideal conditions, each of the sub-carriers in an OFDM signal is periodic within the FFT time buffer. This is critical if the sub-carriers are to remain orthogonal and avoid mutual interference.
A frequency error between the transmitter and the receiver will cause all of the sub-carriers to have a non-integral number of cycles in the FFT time interval, causing leakage. A frequency error shifts the sin(x)/x spectrum of each sub-carrier relative to the FFT frequency bins to the point that the spectral nulls are no longer aligned with the FFT bins. The result is that frequency error causes every sub-carrier to interfere with its neighbors.
Nonlinear Distortion
The effects of nonlinear distortion on an OFDM signal are easily understood by those familiar with inter-modulation distortion. Nonlinear distortion is a particularly important topic for OFDM signals because the signal represents the linear summation of a large number of statistically independent sub-carriers. This results in a signal with Gaussian voltage statistics on the I and Q waveforms.
Sometimes people refer to the signal as having Gaussian statistics in the context of peak-to-average power. This can be misleading because the power statistics are not Gaussian. The I and Q voltage waveforms are Gaussian, but the power, which is the sum of the squares of the I and Q signals, has a chi-square distribution. All that aside, a perfect OFDM signal can have peak envelope power that exceeds the average envelope power by more than 10 dB. This presents the power amplifier designer with some difficult challenges.
In the context of nonlinear distortion, it's safe to model the OFDM signal as a multi-tone signal with each of the tones having a random phase component. In fact, that's exactly what an OFDM signal is. Multi-tone analysis and testing is probably better suited to OFDM signals than to the multi-carrier signals for which it was originally developed.
For most single-carrier modulation formats (excluding code-division multiple access (CDMA) signals), the Nyquist filters. create peak power excursions that occur between symbols.
Errors due to saturation will cause adjacent channel power problems, but may not have a large impact on data transmission—depending on the amount of dispersion in the transmission channel and the receiver's matched filter. For OFDM, the peak power excursions may occur at any time within the symbol interval. Because the OFDM time waveform is a summation of all sub-carriers, nonlinear distortion will create inter-carrier interference. Of course, it will also result in increased adjacent channel power.
Spurious Signals
When a spurious signal is added to an OFDM signal, the effect is similar to what happens with single-carrier modulations. If the spurious tone is at the exact frequency of a sub-carrier, then the effect is to shift the constellation of that sub-carrier—just as carrier leakage shifts the constellation of a single-carrier modulation. Otherwise, the spurious signal causes a spreading of the constellation points for sub-carriers near the spurious tone. Carrier leakage in an OFDM signal only affects the center carrier (if used).
In order to eliminate the effects of the previously described impairments on the OFDM systems, various kinds of compensation circuit and method have been proposed.
U.S. Application Publication No. 20020015450 discloses a method and an arrangement for determining correction parameters used for correcting phase and amplitude imbalance of an I/Q modulator in a transmitter. The transmitter includes an I/Q modulator and a corrector for correcting the phase and amplitude imbalance caused by the I/Q modulator. The arrangement has means for sampling the I/Q-modulated test signal to be transmitted, means for A/D-converting the signal samples taken from the test signal, means for I/Q-demodulating the signal samples digitally into I- and Q-feedback signals, means for determining the phase and amplitude imbalance caused by the I/Q modulator on the basis of the I- and Q-feedback signals, and means for determining the correction parameters of phase and amplitude on the basis of the determined phase and amplitude imbalance.
U.S. Pat. No. 6,208,698 discloses a quadrature modulator controlled by an imbalance estimator, as shown in FIG. 1. The estimator is entirely analog and includes two mixers 120, 121 to which are applied two carrier signals P1 and P2 derived from a carrier signal P obtained from a local oscillator and from a 90° phase-shifter 122. A modulating signal X, Y is applied to each of the mixers 120, 121. The output signals of the mixers 120, 121 are applied to a combiner 123 to constitute a modulated signal S. The modulated signal S is either an intermediate frequency signal or a microwave frequency signal. The estimator 124 of the invention includes means 125 for detecting the instantaneous power Pd of the modulated signal S, means 126, 127 for multiplying the detected instantaneous power Pd by each of the modulating signals X, Y, means 128A, 129A for rectifying the signals produced, which rectifier means can be diodes, for example, and means 130, 131 for integrating the rectified signals, the integrator means being followed by subtractors 160, 161 supplying respective signals E1 and E2 proportional to the amplitude of the modulator imbalance. The subtractors 160 and 161 receive respective analog reference signals REF1 and REF2, which are usually identical (REF1=REF2), the output signals of the integrator means 130, 131 being subtracted from REF1 and REF2, respectively. For example, REF1 and REF2 are equal to twice the mean amplitude of the modulated signal S. The subtractors 160 and 161 generally need to be used only for the purposes of correcting imbalance. The signals E1 and E2 are respectively applied to the multipliers 150 and 151 on the input side of the mixers 121 and 120, respectively, to correct the offsets introduced by these mixers.
Although there are already many kinds of compensation circuit and method, it is still a goal for researchers around the world to propose newer and better solutions to the I/Q imbalance issue in the OFDM systems.